Question

a point t is at (5,1). it is rotated 90 degrees counterclockwise about the origin, then reflected across the line y = - 2, and then translated by (x,y)→(x + 3,y - 1). what are the coordinates of the final image of t? a) (-2,-10) b) (-4,4) c) (2,-8) d) (2,-10)

Explanation:

Step1: Apply 90 - degree counter - clockwise rotation

The rule for a 90 - degree counter - clockwise rotation about the origin for a point $(x,y)$ is $(x,y)\to(-y,x)$. For the point $T(5,1)$, after rotation, it becomes $T_1(-1,5)$.

Step2: Reflect across the line $y = - 2$

The distance between the point $T_1(-1,5)$ and the line $y=-2$ is $d=5-(-2)=7$. The new $y$ - coordinate after reflection across $y = - 2$ is $y=-2 - 7=-9$, and the $x$ - coordinate remains the same. So the point becomes $T_2(-1,-9)$.

Step3: Apply the translation $(x,y)\to(x + 3,y - 1)$

For the point $T_2(-1,-9)$, the new $x$ - coordinate is $x=-1 + 3=2$, and the new $y$ - coordinate is $y=-9-1=-10$. So the final point is $(2,-10)$.

Answer:

D. $(2,-10)$